close form solution for A136010 (per PURRS http://www.cs.unipr.it/purrs/)

[Alexander Povolotsky]

http://www.cs.unipr.it/purrs/
PURRS Demo Results
Exact solution for x(n) = 7*x(-1+n)+9*x(-2+n)
for the initial conditions
x(1) = 20
x(2) = 9

x(n) = -1277/1530*(7/2-1/2*sqrt(85))^n*sqrt(85)    -
131/18*(7/2-1/2*sqrt(85))^n
         +1277/1530 *sqrt(85)*(7/2+1/2*sqrt(85))^n  -
131/18*(7/2+1/2*sqrt(85))^n

for each n >= 1

I checked first few terms and it looks correct.

(11:32) gp > a(n) = -1277/1530*(7/2-1/2*sqrt(85))^n*sqrt(85)-
131/18*(7/2-1/2*sqrt(85))^n+1277/1530
 *sqrt(85)*(7/2+1/2*sqrt(85))^n-131/18*(7/2+1/2*sqrt(85))^n

(11:33) gp > a(1)
%1 = 20.00000000000000000000000000
(11:33) gp > a(2)
%2 = 9.000000000000000000000000000
(11:33) gp > a(3)
%3 = 243.0000000000000000000000000
(11:33) gp > a(4)
%4 = 1782.000000000000000000000001
 --------------------------------------------------------------------------------------------
  On Thu, Mar 27, 2008 at 2:56 PM, Maximilian Hasler
 < Maximilian.Hasler at martinique.univ-ag.fr> wrote:
 Then a g.f. for the sequence including 1st 2 terms would be

   gf = (131*x - 20)/(9*x^2 + 7*x - 1)

  = 20 + 9*x + 243*x^2 + 1782*x^3 + 14661*x^4 + 118665*x^5 +
  962604*x^6 + 7806213*x^7 + 63306927*x^8 + 513404406*x^9 +
  4163593185*x^10 + 33765791949*x^11 + 273832882308*x^12 +
  2220722303697*x^13 + 18009552066651*x^14 + 146053365199830*x^15 +
  O(x^16)

 The denom. being of the form (9x^2+7x-1), this means that
  a[n] = 7a[n-1] + 9a[n-2]

  (in some sense, the higher powers of x add previous terms (coeff's of
  lower powers) to the term corresponding to a given power).
  ---------- Forwarded message ----------
 From: Alexander Povolotsky <apovolot at gmail.com>
 Date: Thu, Mar 27, 2008 at 9:40 AM

  Return-Path: <superseq-reply at research.att.com>

  Report on [ 243,1782,14661,118665,962604,7806213,63306927,513404406]:
  Many tests are carried out, but only potentially useful information
  (if any) is reported here.

  TEST: IS THE SEQUENCE OF ABSOLUTE VALUES IN THE ENCYCLOPEDIA?

  Matches (up to a limit of 50) found for  243 1782 14661 118665 962604
  7806213 63306927 513404406  :

  SUGGESTION: GUESSGF FOUND ONE OR MORE GENERATING FUNCTIONS
  WARNING: THESE MAY BE ONLY APPROXIMATIONS!
  Generating function(s) and type(s) are:

                                 243 + 81 x
                            [- --------------, ogf]
                                  2
                               9 x  - 1 + 7 x

   SUGGESTION: LISTTOALGEQ FOUND ONE OR MORE ALGEBRAIC
  EQUATIONS SATISFIED BY THE GEN. FN.
  WARNING: THESE MAY BE ONLY APPROXIMATIONS!
  Equation(s) and type(s) are:
                                                        2
               [-n + (243 + 7 n) a(n) + (81 + 9 n) a(n) , revogf]


  Types of generating functions that may have been mentioned above:

  ogf     =       ordinary generating function
  egf     =       exponential generating function
  revogf  =       reversion of ordinary generating function
  revegf  =       reversion of exponential generating function
  lgdogf  =       logarithmic derivative of ordinary generating function
  lgdegf  =       logarithmic derivative of exponential generating function


  TRY "GUESSS", HARM DERKSEN'S PROGRAM FOR GUESSING A GENERATING FUNCTION FOR A
  SEQUENCE.

         Guesss - guess a sequence, by Harm Derksen (hderksen at math.mit.edu)

  Guesss suggests that the generating function  F(x)
  may satisfy the following algebraic or differential equation:

  9*x+27+(x^2+7/9*x-1/9)*F(x) = 0

  If this is correct the next 6 numbers in the sequence are:

  [4163593185, 33765791949, 273832882308, 2220722303697, 18009552066651,
  146053365199830]
---------- Forwarded message ----------
From: zak seidov <zakseidov at yahoo.com>
Date: Thu, Mar 27, 2008 at 9:44 PM
Subject: Re: A136010 solved
To: franktaw at netscape.net, djr at nk.ca, seqfan at ext.jussieu.fr,
njas at research.att.com, jordyg365 at gmail.com

And in general, at recurrency
 a[n]==A a[n-1]+B a[n-2], (A>0)

 lim (n->+inf) a(n+1)/a(n)=
 (A+sqrt(A^2+4B))/2.

 --- franktaw at netscape.net wrote:
  > See below for the answer.
 > Franklin T. Adams-Watters
  > -----Original Message-----
 > > From: Don Reble <djr at nk.ca>
 > > Seqfans, njas, Jordan:
 > >
 > > %I A136010
 > > %S A136010
 > > 20,9,243,1782,14661,118665,962604,7806213,63306927,513404406
 > > %N A136010 From an online IQ test (Adaptive IQ ).
 > >    a(1)=20, a(2)=9, a(n+2) = 7*a(n+1) + 9*a(n).
 > >    Quick puzzle: What's lim (n->+inf) a(n+1)/a(n) ?
 > That would be the larger root of x^2-7x-9, which is
 > (7+sqrt(85))/2, or
 > 8.1097722286464436550011371408814....